The proof of th:convalg is done in several steps (see sec:app:proofs for complete derivations). First we prove that almost surely, the sequence of iterates \((w_k)\)  converges to \(C_{\epsilon }\)  (see lm:ineqbound). Then we show that an update step of ASkewSGD  can be written as \(w_{k+1} = w_k - \gamma _k \nabla \ell (w_k) + \gamma _k \eta _{k+1} - \gamma _k u_k\) , where \(\eta _{k+1} = \nabla \ell (w_k) - \widehat{\nabla } \ell (w_k)\)  and \(u_k\)  approximates an element of \(N_{C}(w_k)\) .
We show the convergence of \(\sum _{j=1}^k \gamma _j \eta _{j+1}\)  in lm:martconv, and complete the proof by applying theo:davdrus, which is adapted from [1]}.
